Extending the Kernel for Planar Steiner Tree to the Number of Steiner Vertices

Author Ondrj Suchý



PDF
Thumbnail PDF

File

LIPIcs.IPEC.2015.151.pdf
  • Filesize: 493 kB
  • 12 pages

Document Identifiers

Author Details

Ondrj Suchý

Cite AsGet BibTex

Ondrj Suchý. Extending the Kernel for Planar Steiner Tree to the Number of Steiner Vertices. In 10th International Symposium on Parameterized and Exact Computation (IPEC 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 43, pp. 151-162, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)
https://doi.org/10.4230/LIPIcs.IPEC.2015.151

Abstract

In the Steiner Tree problem one is given an undirected graph, a subset T of its vertices, and an integer k and the question is whether there is a connected subgraph of the given graph containing all the vertices of T and at most k other vertices. The vertices in the subset T are called terminals and the other vertices are called Steiner vertices. Recently, Pilipczuk, Pilipczuk, Sankowski, and van Leeuwen [FOCS 2014] gave a polynomial kernel for Steiner Tree in planar graphs, when parameterized by |T|+k, the total number of vertices in the constructed subgraph. In this paper we present several polynomial time applicable reduction rules for Planar Steiner Tree. In an instance reduced with respect to the presented reduction rules, the number of terminals |T| is at most quadratic in the number of other vertices k in the subgraph. Hence, using and improving the result of Pilipczuk et al., we give a polynomial kernel for Steiner Tree in planar graphs for the parameterization by the number k of Steiner vertices in the solution.
Keywords
  • Steiner Tree
  • polynomial kernel
  • planar graphs
  • polynomial-time preprocessing
  • network sparsification

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads

References

  1. J. Alber, M. R. Fellows, and R. Niedermeier. Polynomial-time data reduction for dominating set. J. ACM, 51(3):363-384, May 2004. Google Scholar
  2. M. W. Bern and P. E. Plassmann. The Steiner problem with edge lengths 1 and 2. Inf. Process. Lett., 32(4):171-176, 1989. Google Scholar
  3. R. van BevernBevern, S. Hartung, F. Kammer, R. Niedermeier, and M. Weller. Linear-time computation of a linear problem kernel for dominating set on planar graphs. In Parameterized and Exact Computation - 6th International Symposium, IPEC 2011, Revised Selected Papers, volume 7112 of LNCS, pages 194-206. Springer, 2011. Google Scholar
  4. A. Björklund, T. Husfeldt, P. Kaski, and M. Koivisto. Fourier meets Möbius: fast subset convolution. In Proceedings of the 39th ACM Symposium on Theory of Computing, STOC 2007, pages 67-74. ACM, 2007. Google Scholar
  5. H. L. Bodlaender, M. Cygan, S. Kratsch, and J. Nederlof. Deterministic single exponential time algorithms for connectivity problems parameterized by treewidth. Inf. Comput., 243:86-111, 2015. Google Scholar
  6. H. L. Bodlaender, R. G. Downey, M. R. Fellows, and D. Hermelin. On problems without polynomial kernels. J. Comput. Syst. Sci., 75(8):423-434, 2009. Google Scholar
  7. G. Borradaile, E. D. Demaine, and S. Tazari. Polynomial-time approximation schemes for subset-connectivity problems in bounded-genus graphs. Algorithmica, 68(2):287-311, 2014. Google Scholar
  8. G. Borradaile, P. N. Klein, and C. Mathieu. An O(n log n) approximation scheme for Steiner tree in planar graphs. ACM Transactions on Algorithms, 5(3), 2009. Google Scholar
  9. J. Byrka, F. Grandoni, T. Rothvoß, and L. Sanità. Steiner tree approximation via iterative randomized rounding. J. ACM, 60(1):6:1-6:33, Feb. 2013. Google Scholar
  10. D. Cieslik. Steiner minimal trees, volume 23. Springer Science & Business Media, 1998. Google Scholar
  11. M. Cygan, H. Dell, D. Lokshtanov, D. Marx, J. Nederlof, Y. Okamoto, R. Paturi, S. Saurabh, and M. Wahlström. On problems as hard as CNF-SAT. In Proceedings of the 27th Conference on Computational Complexity, CCC 2012, pages 74-84. IEEE, 2012. Google Scholar
  12. M. Cygan, J. Nederlof, M. Pilipczuk, M. Pilipczuk, J. M. M. van Rooij, and J. O. Wojtaszczyk. Solving connectivity problems parameterized by treewidth in single exponential time. In IEEE 52nd Annual Symposium on Foundations of Computer Science, FOCS 2011, pages 150-159. IEEE Computer Society, 2011. Google Scholar
  13. M. Dom, D. Lokshtanov, and S. Saurabh. Kernelization lower bounds through colors and IDs. ACM Transactions on Algorithms, 11(2):13:1-13:20, 2014. Google Scholar
  14. R. G. Downey and M. R. Fellows. Parameterized Complexity. Springer-Verlag, New York, 1999. Google Scholar
  15. S. E. Dreyfus and R. A. Wagner. The Steiner problem in graphs. Networks, 1:195-207, 1972. Google Scholar
  16. R. E. Erickson, C. L. Monma, and A. F. Veinott, Jr. Send-and-split method for minimum-concave-cost network flows. Mathematics of Operations Research, 12(4):634-664, 1987. Google Scholar
  17. F. V. Fomin, F. Grandoni, D. Kratsch, D. Lokshtanov, and S. Saurabh. Computing optimal Steiner trees in polynomial space. Algorithmica, 65(3):584-604, 2013. Google Scholar
  18. F. V. Fomin, P. Kaski, D. Lokshtanov, F. Panolan, and S. Saurabh. Parameterized single-exponential time polynomial space algorithm for Steiner tree. In Automata, Languages, and Programming - 42nd International Colloquium, ICALP 2015, Proceedings, Part I, volume 9134 of LNCS, pages 494-505. Springer, 2015. Google Scholar
  19. F. V. Fomin, D. Lokshtanov, and S. Saurabh. Efficient computation of representative sets with applications in parameterized and exact algorithms. In Proceedings of the Twenty-Fifth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2014, pages 142-151. SIAM, 2014. Google Scholar
  20. D. S. R. Frank K. Hwang and P. Winter. The Steiner Tree Problem, volume 53 of Annals of Discrete Mathematics. Elsevier, 1992. Google Scholar
  21. B. Fuchs, W. Kern, D. Mölle, S. Richter, P. Rossmanith, and X. Wang. Dynamic programming for minimum Steiner trees. Theory of Computing Systems, 41(3):493-500, 2007. Google Scholar
  22. M. R. Garey and D. S. Johnson. The rectilinear Steiner tree problem in NP complete. SIAM Journal of Applied Mathematics, 32:826-834, 1977. Google Scholar
  23. M. R. Garey and D. S. Johnson. Computers and Intractability: A Guide to the Theory of NP-Completeness. Freeman, San Francisco, 1979. Google Scholar
  24. S. L. Hakimi. Steiner’s problem in graphs and its implications. Networks, 1:113-133, 1971. Google Scholar
  25. M. Hauptmann and M. Karpinski. A compendium on Steiner tree problems. online. URL: http://theory.cs.uni-bonn.de/info5/steinerkompendium/.
  26. J. Hopcroft and R. Tarjan. Efficient planarity testing. J. ACM, 21(4):549-568, Oct. 1974. Google Scholar
  27. F. K. Hwang, D. S. Richards, and P. Winter. The Steiner Tree Problem. North-Holland, Amsterdam, 1992. Google Scholar
  28. M. Jones, D. Lokshtanov, M. S. Ramanujan, S. Saurabh, and O. Suchý. Parameterized complexity of directed Steiner tree on sparse graphs. In Algorithms - ESA 2013 - 21st Annual European Symposium, Proceedings, volume 8125 of LNCS, pages 671-682. Springer, 2013. Google Scholar
  29. A. B. Kahng and G. Robins. On Optimal Interconnections for VLSI. Kluwer Academic Publisher, 1995. Google Scholar
  30. P. Klein and R. Ravi. A nearly best-possible approximation algorithm for node-weighted Steiner trees. Journal of Algorithms, 19(1):104 - 115, 1995. Google Scholar
  31. B. Korte, H. J. Prömel, and A. Steger. Steiner trees in VLSI-layout. In Paths, Flows and VLSI-Layout, pages 185-214, 1990. Google Scholar
  32. A. Y. Levin. Algorithm for the shortest connection of a group of graph vertices. Sov. Math. Dokl., 12:1477-1481, 1971. Google Scholar
  33. J. Nederlof. Fast polynomial-space algorithms using inclusion-exclusion. Algorithmica, 65(4):868-884, 2013. Google Scholar
  34. M. Patrascu and R. Williams. On the possibility of faster SAT algorithms. In Proceedings of the Twenty-First Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2010, pages 1065-1075. SIAM, 2010. Google Scholar
  35. M. Pilipczuk, M. Pilipczuk, P. Sankowski, and E. J. van Leeuwen. Subexponential-time parameterized algorithm for Steiner tree on planar graphs. In 30th International Symposium on Theoretical Aspects of Computer Science, STACS 2013, volume 20 of LIPIcs, pages 353-364. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, 2013. Google Scholar
  36. M. Pilipczuk, M. Pilipczuk, P. Sankowski, and E. J. van Leeuwen. Network sparsification for Steiner problems on planar and bounded-genus graphs. In 55th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2014, pages 276-285. IEEE Computer Society, 2014. Google Scholar
  37. H. J. Prömel and A. Steger. The Steiner Tree Problem; a Tour through Graphs, Algorithms, and Complexity. Vieweg, 2002. Google Scholar
Questions / Remarks / Feedback
X

Feedback for Dagstuhl Publishing


Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail